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Title Page 2 Copyright 3 Contents 4 Introduction 10 Chapter 1 Introduction to Arbitrary Lagrangian?Eulerian in Finite Element Methods 17 1.1. Introduction 17 1.2. Governing equations 19 1.3. Operator splitting 20 1.4. The Lagrangian step 23 1.4.1. Governing equations 23 1.4.2. The central difference method 24 1.4.3. Element formulation 27 1.4.4. Hourglass modes 30 1.4.5. Stress rates 32 1.4.6. Shock viscosity 34 1.4.6.1. von Neumann?Richtmyer 35 1.4.6.2. Standard quadratic and linear formulation 36 1.4.6.3. Effect on time step size 37 1.4.7. Mixture theories 38 1.4.7.1. Mean strain rate mixture theory 38 1.4.7.2. Mean stress mixture theory 40 1.5. Mesh relaxation 41 1.6. The Eulerian step 43 1.6.1. Transport in one dimension 45 1.6.2. Multidimensional transport by operator splitting 48 1.6.3. Multidimensional transport on unstructured meshes 50 1.6.4. Momentum transport 51 1.6.4.1. Momentum transport using a dual mesh in one dimension 52 1.6.4.2. Element-centered transport in one dimension SALE [AMD 80] introduced the first element-centered transport method. In one dimension, the momentum density and element momentum are defined as 53 1.6.5. Interface reconstruction 55 1.6.5.1. Lag

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4<8=8AB@0B>@

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